Supposedly it is now proven. I do not get this at all, here are some questions:

1. Why do we set a and b to 2n and 2m?2. Why the 2 at all? 3. What exactly is m and n?4. How does 2(n + m) prove this question true at all? For example, how can I do a check using the proof to see some true outcomes?

To express the numbers in a generic way, but in a way that shows that the numbers are even, not odd. This is the same process that you used back in beginning algebra when you did "number" word problems.

The definition of an "even" number is that it is a multiple of two. Therefore, the generic form of an even number is the product of two and some variable. To show that the sum is also even, it must be shown that the sum is a multiple of two.